1.We showed an asymptotic second-order efficiency of a two-stage procedure which was proposed to construct a fixed-width confidence interval for a linear function of normal means.2.We showed an asymptotic second-order efficiency of a two-stage estimation for a fixed-span confidence region about a linear function of normal mean vectors.3.We considered multistage estimation procedures to achieve a bounded risk when estimating the mean of a normal distribution under a LINEX loss function. Two-stage and three-stage estimation procedures are introduced and their second-order asymptotic was examined.4.We proposed a two-stage estimation for a fixed-span confidence region about a linear function of mean vectors of multivariate normal distributions when the covariance matrices have some structures. The procedure was shown to be asymptotically second-order efficient.5.We considered a three-stage procedure which was proposed to yield a fixed-width confidence interval of the normal mean with a precise confidence level. The procedure was shown to be asymptotically second-order efficient.6.We considered a sequential point estimation procedure to achieve a bounded risk when estimating a linear function of normal means under a LINEX loss function. A two-stage estimation procedure was introduced to achieve the goal and its asymptotic property was examined.